We address the problem of identifying underground anomalies (e.g. holes) based on gravity measurements. This is a theoretically well-studied yet diffi cult problem. In all except a few special cases, the inverse problem has multiple solutions, and additional constraints are needed to regularize it. Our approach makes general assumptions about the shape of the anomaly that can also be seen as sparsity assumptions. We can then adapt recently developed sparse reconstruction algorithms to address this problem. The results are extremely promising, even though the theoretical assumptions underlying sparse recovery do not hold for gravity problems of this kind. We examine several types of sparse bases in the context of this gravity inverse problem and compare and contrast their relative merits.